Initial successive coefficients for certain classes of univalent functions involving the exponential function

Initial successive coefficients for certain classes of univalent functions

We consider a family of all analytic and univalent functions in the unit disk of the form $f(z)=z+a_2z^2+a_3z^3+\cdots$. The aim of this article is to investigate the bounds of the difference of moduli of initial successive coefficients, i.e. $\big | |a_{n+1}|-|a_n|\big |$ for $n=1,\,2$ and for some subclasses of analytic univalent functions. We found that all the estimations are sharp in nature by constructing some extremal functions.

On the successive coefficients of certain univalent functions

The object of this paper is to study relationship between successive coefficients of some subclasses of the class of univalent functions in the unit disk. the result obtained is sharp, and is used to provide a new, short proof of the well-known conjecture of Robertson on the coefficients of close-to-convex functions.

Initial Successive coefficients of Inverse functions of certain classes of univalent functions

We consider functions of the type $f(z)=z+a_2z^2+a_3z^3+\cdots$ from a family of all analytic and univalent functions in the unit disk. Let $F$ be the inverse function of $f$, given by $F(z)=w+\sum_{n=2}^{\infty}A_nw^n$ defined on some $|w|\le r_0(f)$. In this paper, we find the sharp bounds of $\big | |A_{n+1}|-|A_n|\big |$, for $n=1,\,2$, for some subclasses of univalent functions.

单值函数的复指数积分均值谱与布伦南猜想

Complex exponential integral means spectra of univalent functions and the Brennan conjecture

本文研究单值函数在单位圆内复指数积分均值谱的连续性，证明了其与布伦南猜想的相关性，并确定了所有单值有理函数的积分均值谱。

与指数映射相关的单叶函数子类的凹性半径

Radii of Concavity for Subclasses of Univalent Functions Associated with the Exponential Mapping

本文在Avkhadiev--Wirths几何框架下，通过显式超越方程刻画了与指数映射相关的单叶函数类的凹性半径，构造极值函数证明全局尖锐性，并数值验证了半径关于参数A的严格单调性。

Region of variability for exponentially convex univalent functions

For $α\in\IC\setminus \{0\}$ let $\mathcal{E}(α)$ denote the class of all univalent functions $f$ in the unit disk $\mathbb{D}$ and is given by $f(z)=z+a_2z^2+a_3z^3+\cdots$, satisfying $$ {\rm Re\,} \left (1+ \frac{zf''(z)}{f'(z)}+αzf'(z)\right)>0 \quad {in ${\mathbb D}$}. $$ For any fixed $z_0$ in the unit disk $\mathbb{D}$ and $λ\in\overline{\mathbb{D}}$, we determine the region of variability $V(z_0,λ)$ for $\log f'(z_0)+αf(z_0)$ when $f$ ranges over the class $$\mathcal{F}_α(λ)=\left\{f\in\mathcal{E}(α) \colon f''(0)=2λ-α%\quad{and} f'''(0)=2[(1-|λ|^2)a+ %(λ-α)^2 -λα] \right\}. $$ We geom…